In the special case where the graph is unweighted, the \(W\) -polynomial is known as the \(U\) -polynomial. The \(U\) -polynomial, Brylawski's polychromate [1]} and a generalisation of Stanley's chromatic symmetric function known as the Tutte symmetric function [2]} have been shown to be equivalent in [3]}, [4]}. Because the combinatorial Laplacian characteristic polynomial of an unweighted graph is, up to a sign, an evaluation of the \(U\) -polynomial, results on the three equivalent graph polynomials can give results on the combinatorial Laplacian.
